Diffusion with Localized Traps: Mean
First Passage Time, Eigenvalue
Asymptotics, and Fekete Points
D. Coombs (UBC); R. Straube (Magdeburg); A. Peirce (UBC), S. Pillay (UBC)
T. Kolokolnikov (Dalhousie); A. Cheviakov (Saskatchewan); P. Bressloff (Utah)
M.J. Ward (UBC)
ward@math.ubc.ca
GWU – p.1
Outline of the Talk
Some General Considerations:
1. Diffusion with Localized Traps (Biological Motivation; from a
Mathematician’s Viewpoint). The Narrow Escape Problem.
2. Eigenvalue Problems in Perforated Domains and in Domains with
Perforated Boundaries. Eigenvalue Optimization and the Mean First
Passage Time (General)
3. Fekete Points
Specific Problems Considered:
1. Eigenvalue Asymptotics in 2-D or 3-D Perforated Domains.
2. Diffusion on the Surface of a Sphere
3. The Mean First Passage Time for Escape from a Sphere
GWU – p.2
Narrow Escape Problem I
Brownian motion with diffusivity D in Ω with ∂Ω insulated
except for an (multi-connected) absorbing patch ∂Ωa of measure O(ε). Let
∂Ωa → xj as ε → 0 and X(0) = x ∈ Ω be initial point for Brownian motion.
Narrow Escape:
The MFPT v(x) = E [τ |X(0) = x] satisfies (Z. Schuss (1980))
1
∆v = − , x ∈ Ω ,
D
∂n v = 0 x ∈ ∂Ωr ; v = 0 ,
x ∈ ∂Ωa = ∪N
j=1 ∂Ωεj .
GWU – p.3
Narrow Escape Problem II
Key General References:
Z. Schuss, A. Singer, D. Holcman, The Narrow Escape Problem for
Diffusion in Cellular Microdomains, PNAS, 104, No. 41, (2007),
pp. 16098-16103.
O. Bénichou, R. Voituriez, Narrow Escape Time Problem: Time
Needed for a Particle to Exit a Confining Domain Through a Small
Window, Phys. Rev. Lett, 100, (2008), 168105.
S. Condamin, et al., Naure, 450, 77, (2007)
Relevance of Narror Escape Time Problem in Biology:
time needed for a reactive particle released from a specific organelle
to activate a given protein on the cell membrane
biochemical reactions in cellular microdomains, like dendritic spines,
synapses, or microvesicles. Such submicron domains often contain a
small amount of particles that must first exit domain to fullfill a
biological function.
GWU – p.4
Diffusion of Protein Receptors: I
Diffusion of protein receptors on a cylindrical dendritic membrane
Ω = {|x| < L, |y| < 2πl}, with partially absorbing traps.
(A)
Uj
(C)
EXO
σ0
ωj
Rj
END
DEG
receptor
Sj
x=0
δj
dendrite
kj
σREC
j
spine
neck
σDEG
j
spine
l
x=L
(B)
y = πl
Ωε
Ωj
x=0
y = -πl
x=L
GWU – p.5
Diffusion of Protein Receptors: II
Model: Localized Traps
and σ > 0 is protein receptors influx from the soma:
Ut = ∆U ,
∂x U (−L, y) = −σ ,
x ∈ Ω\Ωp ,
∂x U (L, y) = 0 ;
ε∂n U = −κj (U − Tj ) ,
Ωp = ∪N
j=1 Ωεj ,
U , ∂y U ,
x ∈ ∂Ωεj ,
2πl periodic in y ,
j = 1, . . . , N .
Define the average concentration Uj on the j th spine boundary
Z
1
Uj =
U dx .
2πε ∂Ωε
j
Within each spine Tj (t) and Sj (t) for j = 1, . . . , N satisfy coupled ODE’s
0
Tj = Fj (Tj , Sj , Uj ) ,
0
Sj = Hj (Tj , Sj ) .
Model due to Bressloff and Earnshaw (Phys. Rev. E. (2007), J. Neuroscience
(2006)). The 1-D steady-state problem studied.
steady state problem studied in Bressloff, Earnshaw, MJW, SIAP
(2008).
2-D
GWU – p.6
Cell Signalling From Small Compartments
Spatial gradients of activated signalling
molecules from small compartments inside a cell. Stationary
concentration for fraction c = ca /ct of such molecules:
Model of Straube, MJW (SIAP, 2009):
∂n c = 0 , x ∈ ∂Ω
∆c − α2 c = 0 , x ∈ Ω\∪N
j=1 Ωεj ;
(
saturated enzyme ,
σj , x ∈ ∂Ωεj
ε∂n c =
κj (1 − c) , x ∈ ∂Ωεj un-saturated enzyme ,
B Kholodenko, Cell-Signalling Dynamics in Time and Space, Nat
Rev Mol Cell Biol, (2006).
General:
GWU – p.7
Eigenvalues in Perforated Domains I
For a bounded 2-D or 3-D domain;
∆u + λu = 0 ,
∂n u = 0
x ∈ Ω\Ωp ;
x ∈ ∂Ω ,
Z
u2 dx = 1 ,
Ω\Ωp
u = 0,
x ∈ ∂Ωp .
Here Ωp = ∪N
i=1 Ωεi are N interior non-overlapping holes or traps, each
of ‘radius’ O(ε) 1.
Also Ωεi → xi as ε → 0, for i = 1, . . . , N . The centers xi are arbitrary.
wandering particle
O(ε )
x2
N small absorbing holes
x1
n
reflecting
walls
GWU – p.8
Eigenvalues in Perforated Domains II
Eigenvalue Asymptotics for Principal Eigenvalue λ1 :
For the case of N circular holes each of radius
ε 1, Ozawa (Duke J., 1981) proved that
Previous Studies in 2-D:
2πN ν
λ1 ∼
+ O(ν 2 ),
|Ω|
1
1.
ν≡−
log ε
For the case of N localized traps, Ozawa (J. Fac.
Soc. U. Tokyo, 1983) (see also Flucher (1993)) proved that
Previous Studies in 3-D:
N
4πε X
Cj + 0(ε2 ) .
λ1 ∼
|Ω| j=1
Here Cj is the electrostatic capacitance of the j th trap defined by
∆y w = 0 ,
w = 1,
Remark:
y ∈ ∂Ωj ;
y 6∈ Ωj ≡ ε−1 Ωεj ,
w∼
Cj
,
|y|
problem dates back to Szego 1930’s.
|y| → ∞ .
GWU – p.9
Eigenvalues in Perforated Domains III
The Mean First Passage Time v(x) for diffusion in a perforated
domain with initial starting point x ∈ Ω\Ωp satisfies (ref. Z. Schuss, (1980))
The MFPT:
1
, x ∈ Ω\Ωp ;
D
∂n v = 0 x ∈ ∂Ω , v = 0 ,
∆v = −
x ∈ ∂Ωp .
Relationship Between Averaged MFPT and Principal Eigenvalue:
v̄ ≡ χ ∼
1
,
Dλ1
v̄ ≡
1
|Ω|
Z
is that for ε → 0
v dx
Ω
Let λ1 > 0 be the fundamental eigenvalue. For ε → 0 (small hole
radius) find the hole locations xi , for i = 1, . . . , N , that maximize λ1 .
Goal:
In other words, chose the trap locations to minimize the lifetime of a
wandering particle in Ω. Maximizing λ1 is equivalent to minimizing v̄.
Goal:
Extend planar 2-D case to a manifold; surface of a sphere.
Since the previous results for λ1 are independent of trap
locations xj , j = 1, . . . , N , we need higher order terms to optimize λ1 .
Key Point:
GWU – p.10
Eigenvalues and Narrow Escape I
For ε → 0, v̄ ∼ 1/(Dλ1 ), where λ1 is the first eigenvalue of
Z
u2 dx = 1 ,
∆u + λu = 0 , x ∈ Ω ;
Ω
∂n u = 0
x ∈ ∂Ωr ,
u = 0,
x ∈ ∂Ωa = ∪N
j=1 ∂Ωεj .
For a 2-D domain with smooth boundary (MJW, Keller, SIAP, 1993)
πN ν
+ O(ν 2 ),
λ1 ∼
|Ω|
1
1.
ν≡−
log ε
For a 3-D domain with smooth boundary (MJW, Keller, SIAP, 1993)
N
2πε X
Cj + 0(ε2 ) .
λ1 ∼
|Ω| j=1
Here Cj is the capacitance of the electrified disk problem
∆y w = 0 ,
w = 1,
y3 ≥ 0,
−∞ < y1 , y2 < ∞ ,
y3 = 0 , (y1 , y2 ) ∈ ∂Ωj ; ∂y3 w = 0 ,
w ∼ Cj /|y| ,
y3 = 0 , (y1 , y2 ) ∈
/ ∂Ωj ;
|y| → ∞ .
GWU – p.11
Eigenvalues and Narrow Escape II
SOME RECENT WORK IN 2-D
For ε → 0,
|Ω|
[− log ε + O(1)] .
πD
Ref: D. Holcman, Z. Schuss, J. Stat. Phys., 117, (2004), pp. 975–1014.
v(x) =
For the unit disk, with x1 = (1, 0)
v(0) = E [τ |X(0) = 0] ∼
1
|Ω|
− log ε + log 2 +
.
πD
4
A. Singer, Z. Schuss, D. Holcman, J. Stat. Phys. 122, (2006), pp.
465-489.
Ref:
Analysis of v(x) for two traps on the unit disk or unit sphere (up to
undertermined O(1) terms fit through Brownian particle simulations).
Ref: D. Holcman, Z. Schuss, J. of Phys. A: Math Theor., 41, (2008),
155001.
GWU – p.12
Eigenvalues and Narrow Escape III
SOME RECENT WORK IN 3-D
For one circular trap of radius ε on the unit sphere Ω with |Ω| = 4π/3,
i
ε
|Ω| h
1 − log ε + O (ε) ,
v̄ ∼
4εD
π
Ref:
122,
A. Singer, Z. Schuss, D. Holcman, R. S. Eisenberg, J. Stat. Phys.,
No. 3, (2006), pp. 437–463.
For arbitrary Ω with smooth boundary and one circular trap
i
ε
|Ω| h
1 − H log ε + O (ε) .
v̄ ∼
4εD
π
Here H is the mean curvature of ∂Ω at the center of the circular trap.
Ref: A. Singer, Z. Schuss, D. Holcman, Phys. Rev. E., 78, No. 5,
051111, (2009).
GWU – p.13
Eigenvalues and Fekete Points: I
Calculate higher-order expansions for v(x) and v̄ as ε → 0 in
2-D and 3-D to determine the signficant effect on v̄ of the spatial
configuration {x1 , · · · , xN } of absorbing boundary traps for a fixed area
fraction of traps. Optimize v̄ with respect to {x1 , · · · , xN }.
Main Goal:
One Specific Question in 3-D:
Let Ω be the unit sphere with N -circular absorbing patches on ∂Ω of a
common radius. Is minimizing v̄ equivalent to minimizing the discrete
energy Hc (x1 , . . . , xN ) defined by
Hc (x1 , . . . , xN ) =
N X
N
X
j=1
k=1
1
,
|xj − xk |
|xj | = 1 .
k6=j
Such points are Fekete points. They correspond to finding the minimal
discrete energy of “electrons” confined to the boundary of a sphere.
(Discovery of Carbon-60 molecules. Long list of references;
Thompson, E. Saff, N. Sloane, A. Kuijlaars etc..) For narrow escape
from a sphere, we show that a specific generalization of this discrete
energy is central.
GWU – p.14
Eigenvalues and Fekete Points: II
Specific Question in 2-D:
correspond to the minimum point of the logarithmic
energy HL on the unit sphere
Elliptic Fekete points:
HL (x1 , . . . , xN ) = −
N X
N
X
j=1
log |xj − xk | ,
|xj | = 1 .
k=1
k6=j
(Long list of references; Smale and Schub, Saff, Sloane, Kuijlaars,...)
Is there a connection between these points and perturbed eigenvalue
problems with traps? Yes, for diffusion on the surface of a sphere with
traps.
GWU – p.15
Eigenvalues in 2-D Perforated Domains: I
T. Kolokolnikov, M. Titcombe, MJW,
“Optimizing the Fundamental Neumann Eigenvalue for the Laplacian in a
Domain with Small Traps”, EJAM Vol. 16, No. 2, (2005), pp. 161-200.
Eigenvalue Optimization in 2-D:
Key Quantity:
Neumann G-function Gm (x; x0 ), and regular part Rm (x; x0 ):
1
− δ(x − x0 ) , x ∈ Ω ,
|Ω|
Z
= 0 , x ∈ ∂Ω ;
Gm dx = 0 ,
∆Gm =
∂ n Gm
Ω
1
log |x − x0 | + Rm (x, x0 ) .
Gm (x, x0 ) = −
2π
The Green’s matrix G is defined in terms of the interaction term
Gm (xi ; xj ) ≡ Gmij , and the self-interaction Rm (xi ; xi ) ≡ Rmii by


···
···
Gm1N
Rm11 Gm12


···
··· 
 Gm21 Rm22 Gm23
.
G≡
..
..
..
..
..


.


.
.
.
.
GmN 1
···
···
GmN N −1
RmN N
GWU – p.16
Eigenvalues in 2-D Perforated Domains: II
For N small holes centered at x1 , . . . , xN with
logarithmic capacitances d1 , . . . , dN , then
Principal Result (KTW):
N
N N
2π X
4π 2 X X
νj −
νj νk (G)jk + O(ν 3 ) .
λ1 (ε) ∼
|Ω| j=1
|Ω| j=1
k=1
Here νj ≡= −1/ log(εdj ) and (G)jk are the entries of G-matrix G. For N
circular holes of a common radius ε, then dj = 1, ν = −1/ log ε, and
4π 2 ν 2
2πN ν
−
p(x1 , . . . , xN ) + O(ν 3 ) ,
λ1 (ε) ∼
|Ω|
|Ω|
Note: the logarithmic capacitance dj of the j th hole is defined by
∆y v = 0 ,
v = 0,
y 6∈ Ωj ≡ ε−1 Ωεj ,
y ∈ ∂Ωj ,
v ∼ log |y| − log dj + o(1) ,
|y| → ∞ .
It can be calculated analytically for ellipses, two closely spaced circular
disks, etc.
GWU – p.17
Eigenvalues in 2-D Perforated Domains: III
Discrete Sum:
The discrete sum p(x1 , . . . , xN ) is defined by
p(x1 , . . . , xN ) ≡
N X
N
X
(G)jk .
j=1 k=1
For N circular holes of radius ε 1, λ1 has a local maximum at
a local minimum point of the “Energy-like” function p(x1 , . . . , xN ).
Key Point:
Specific Questions Adressed in [KTW]:
For N = 1 (one hole), then p = Rm (x1 , x1 ). Can we find domains Ω
where there are there several points x1 that locally maximize λ1 .
Multiplicity of critical points of Rm ? (Yes, for a class of dumbell-shaped
domains).
For the unit disk Ω = |x| ≤ 1, determine ring-type configurations of
holes x1 , ..., xN that maximize λ1 .
GWU – p.18
Eigenvalues in 2-D Perforated Domains: IV
Multiple Holes in the Unit Disk:
and Rm are
Let Ω be the unit disk with |Ω| = π. Then, Gm
1
log |x − ξ| + Rm (x; ξ)
2π
(|x|2 + |ξ|2 ) 3
1
ξ
+
log x|ξ| −
− .
Rm (x; ξ) = −
2π
|ξ|
2
4
Gm (x; ξ) = −
For the unit disk, minimizing p(x1 , . . . , xN ) is equivalent to the minimizing
F(x1 , . . . , xN ) for |xj | < 1 where
F(x1 , . . . , xN ) = −
N X
N
X
j=1
k=1
k6=j
log |xj − xk | −
N X
N
X
j=1 k=1
log |1 − xj x̄k | + N
N
X
|xj |2 .
j=1
For the GL model of superconductivity in the unit disk, equilibrium
vortices at x1 , . . . , xN with |xj | < 1 and a common winding number are
located at critical points of F without confining potential term.
GWU – p.19
Eigenvalues in 2-D Perforated Domains: V
Optimize F over certain ring-type configurations of
holes. We then compare the results with those computed with
optimization software from MATLAB.
Restricted Optimization:
Two Patterns: I (one ring), II (ring with a center hole). Specifically,
xj = re2πij/N ,
j = 1, . . . , N ,
xj = re2πij/(N −1) ,
(P I) ,
j = 1, . . . , N − 1 , xN = 0 ,
(P II) .
More generally, construct m ring patterns with equidistantly spaced
traps on each ring. Paramaters are the ring radii r1 , . . . , rm , the number
of traps on each ring, and the phase angle relative to each ring.
For each pattern we can calculate p(x1 , . . . , xN ) explicitly and then
optimize over the ring radii.
.
GWU – p.20
Eigenvalues in 2-D Perforated Domains: VI
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
GWU – p.21
Eigenvalues in 3-D Perforated Domains: I
In a 3-D bounded domain Ω consider
∆u + λu = 0 ,
∂n u = 0
x ∈ Ω\Ωp ;
x ∈ ∂Ω ,
Z
u2 dx = 1 ,
Ω\Ωp
u = 0,
x ∈ ∂Ωp .
Here Ωp = ∪N
i=1 Ωεi , with Ωεi → xi as ε → 0 and non-overlapping.
For N small traps centered at x1 , . . . , xN
with capacitances C1 , . . . , CN , then
Principal Result (Cheviakov, MJW):
N
N N
4πε X
16π 2 ε2 X X
Cj −
Cj Ck (G)jk + O(ε3 ) .
λ1 ∼
|Ω| j=1
|Ω| j=1
k=1
Here (G)jk ≡ Gm (xj ; xk ) for j 6= k and (G)jj ≡ Rm (xj ; xj ) where Gm (x; ξ)
and Rm (x; ξ) are now the 3-D Neumann G-function for the Laplacian.
GWU – p.22
Eigenvalues in 3-D Perforated Domains: II
The matrix G can be found explicitly when Ω is the unit sphere. By
summing series related to Legendre polynomials
2
1
1
1
Gm (x; ξ) =
+
ln
+
4π|x − ξ| 4π|x|r 0
4π
1 − |x||ξ| cos θ + |x|r 0
7
1
2
2
.
|x| + |ξ| −
+
8π
10π
Here r 0 = |x0 − ξ|, where x0 = x/|x|2 is the image point and θ is the angle
between x and ξ. The regular part Rm (ξ, ξ) is
2
1
7
1
|ξ|
2
−
log 1 − |ξ| +
−
.
Rm (ξ, ξ) =
4π (1 − |ξ|2 ) 4π
4π
10π
Open Problems:
Where are the optimal trap locations xj for j = 1, . . . , N inside the unit
sphere that maximize the first eigenvalue? For identical traps
P Pwe need
Gjk .
to minimize the explicitly known function p(x1 , . . . , xN ) =
What about more general domains, such as a cube? Here we need
Ewald summation techniques to build the matrix G.
GWU – p.23
Diffusion on the Surface of a Sphere: I
The surface diffusion problem is formulated as
4s u = −M ,
x ∈ Sε ≡ S\ ∪N
j=1 Ωεj ,
ε∇s u · n̂ + κj (u − bj ) = 0 ,
x ∈ ∂Ωεj .
S is the unit sphere, Ωεj are localized circular traps of radius O(ε) on S
centered at xj with |xj | = 1 for j = 1, . . . , N .
Traps are non-overlapping; ∆s is surface Laplacian.
GWU – p.24
Diffusion on the Surface of a Sphere: II
When M = −1/D, u is the Mean First Passage Time (MFPT)
for diffusion on S with diffusivity D (Z. Schuss).
Problem 1:
Problem 2:
u is concentration and M (θ, φ) arises from processes inside S.
Construct the asymptotic solution for u in the limit of small trap radii
ε → 0 for both problems. We focus on Problem 1.
Goal:
Eigenvalue Problem:
The corresponding eigenvalue problem on S is
4s ψ + σψ = 0 ,
x ∈ Sε ≡ S\ ∪N
j=1 Ωεj ,
ε∇s ψ · n̂ + κj ψ = 0 , x ∈ ∂Ωεj ,
Z
ψ 2 ds = 1 .
S
Calculate the principal eigenvalue σ1 in the limit ε → 0. This
determines the rate of approach to the steady-state.
Goal:
D. Coombs, R. Straube, MJW, “Diffusion on a Sphere with
Traps...”, to appear, SIAM (2009).
Reference:
GWU – p.25
Diffusion on the Surface of a Sphere: III
For one perfectly absorbing trap at the north pole
with M = 1/D, we get an ODE problem for u(θ):
Previous Results for MFPT:
1
1 ∂
(sin θ ∂θ u) = − ,
sin θ ∂θ
D
θc < θ < π ; u(θc ) = 0 , u0 (π) = 0 .
The solution with θc = ε 1 is
ε
i
1 h
−2 log
+ log(1 − cos θ) ,
u∼
D
2
ε
i
1 h
ū ∼
−2 log
−1 .
D
2
Lindeman, Laufenberger, Biophys. (1986); Singer et al. J. Stat. Phys.
(2006).
Ref:
Previous Results for Principal Eigenvalue:
near the south pole, i.e. θc = π − ε,
∂θθ ψ + cot(θ)∂θ ψ + σψ = 0 ,
For one perfectly absorbing trap
0 < θ < θc ;
ψ(θc ) = 0 , ψ 0 (0) = 0 .
An explicit solution (Weaver (1983), Chao et. al. (1981), Biophys. J.) gives
1
µ
log 2 1
+
; µ=−
σ ∼ + µ2 −
2
2
4
log ε
GWU – p.26
Diffusion on the Surface of a Sphere: IV
Problem 1 (MFPT):
analysis yields
Let M be constant and bj = 0. A matched asymptotic
Principal Result:
Consider N partially absorbing circular traps of radii
εaj 1 centered at xj , for j = 1, . . . , N on S. Then, the asymptotics for u
in the “outer” region |x − xj | O(ε) for j = 1, . . . , N is
u(x) = −2π
N
X
j=1
Aj G(x; xj ) + χ ,
χ≡
1
4π
Z
u ds ,
S
where Aj for j = 1, . . . , N has the asymptotics with logarithmic gauge µj


N
X
2M µj 
2

pw (x1 , . . . , xN ) + O(|µ|2 ) .
Aj =
µi log |xi − xj | −
1 +
N µ̄
N µ̄
j=1
j6=i
The averaged MFPT ū = χ is given asymptotically by
4
2M
+ M (2 log 2 − 1) − 2 2 pw (x1 , . . . , xN ) + O(|µ|) .
ū = χ =
N µ̄
N µ̄
GWU – p.27
Diffusion on the Surface of a Sphere: V
Here µj , µ̄, and the weighted discrete energy pw (x1 , . . . , xN ), are
1
,
µj ≡ −
log(εβj )
βj ≡ aj exp (−1/aj κj ) ;
pw (x1 , . . . , xN ) ≡
N X
N
X
N
1 X
µ̄ ≡
µj ;
N j=1
µi µj log |xi − xj | .
i=1 j>i
The Green’s function G(x; x0 ) that appears satisfies
Z
1
− δ(x − x0 ) , x ∈ S ;
G ds = 0
4s G =
4π
S
G is 2π periodic in φ and smooth at θ = 0, π .
It is given analytically by
1
log |x − x0 | + R ,
G(x; x0 ) = −
2π
1
[2 log 2 − 1] .
R≡
4π
G appears in various studies of the motion of fluid vortices on S
(P. Newton, S. Boatto, etc..).
Remark:
GWU – p.28
Diffusion on the Surface of a Sphere: VI
Principal Result:
For N identical perfectly absorbing traps of a common
radius εa centered at xj , for j = 1, . . . , N , on S, the principal eigenvalue
has asymptotics
2
µN
N
+ µ2 −
(2 log 2 − 1) + p(x1 , . . . , xN ) + O(µ3 ) ,
σ(ε) ∼
2
4
where p(x1 , . . . , xN ) is the discrete logarithmic energy and µ is
p(x1 , . . . , xN ) ≡
N X
N
X
log |xi − xj | . ,
µ≡−
i=1 j>i
1
log(εa)
For N = 1, we get (in agreement with old results)
µ µ2
(1 − 2 log 2) .
σ(ε) ∼ +
2
4
Key Point:
σ(ε) is maximized at the elliptic Fekete points.
Can formulate a problem involving the Helmholtz Green’s
function on the sphere that sums the infinite logarithmic expansion for
σ(ε). Result above has error of O(µ3 ).
Remark:
GWU – p.29
Diffusion on the Surface of a Sphere: VII
Summing the infinite logarithmic series for σ(ε) yields:
Principal Result:
Consider N partially absorbing traps of radii εaj for
j = 1, . . . , N . Let ν(ε) be the smallest root of the transcendental equation
Det (I + 2πRh U + 2πGh U) = 0 .
Here U is the diagonal matrix with Ujj = µj for j = 1, . . . , N , and Gh is the
Helmholtz Green’s function matrix with matrix entries
1
|xj − xi |2
Pν
− 1 , i 6= j ,
Ghij = −
Ghjj = 0 ;
4 sin(πν)
2
Then, with an error of order O(ε), σ(ε) ∼ ν(ν + 1).
Pν (z) is the Legendre function of the first kind, with regular part
Rh (ν) ≡ −
1
[−2 log 2 + 2γ + 2ψ(ν + 1) + π cot(πν)] .
4π
γ is Euler’s constant, ψ is Di-gamma function, and recall
1
,
µj ≡ −
log(εβj )
βj ≡ aj exp (−1/aj κj ) ;
GWU – p.30
Diffusion on the Surface of a Sphere: VIII
Table 1: Smallest eigenvalue σ(ε) for the 2- and 5-trap configurations. For
the 2-trap case the traps are at (θ1 , φ1 ) = (π/4, 0) and (θ2 , φ2 ) = (3π/4, 0).
Here, σ is the numerical solution found by COMSOL; σ ∗ corresponds to
summing the log expansion; σ2 is calculated from the two-term expansion.
5 traps
2 traps
σ
σ∗
σ2
σ
σ∗
σ2
0.02
0.7918
0.7894
0.7701
0.2458
0.2451
0.2530
0.05
1.1003
1.0991
1.0581
0.3124
0.3121
0.3294
0.1
1.5501
1.5452
1.4641
0.3913
0.3903
0.4268
0.2
2.5380
2.4779
2.3278
0.5177
0.5110
0.6060
ε
For ε = 0.2 and N = 5, we get 5% trap surface area fraction. The
agreement is very good.
Note:
GWU – p.31
Diffusion on the Surface of a Sphere: IX
Effect of Spatial Arrangement of Traps:
2.5
2.0
4
4
1.5
4
σ
4
4
4
4
1.0
4
2.0
4
4
4
χ
4
4
4
1.0
0.5
0.0
0.00
1.5
4
4
4
4
0.5
0.05
0.10
0.15
0.20
0.25
0.0
0.00
0.05
0.10
0.15
4
4
4
0.20
4
4
4
0.25
Note: ε = 0.1 corresponds to 1% trap surface area fraction.
Plots: Results for σ(ε) (left) and χ(ε) (right) for three different
4-trap
patterns with perfectly absorbing traps and a common radius ε. Heavy
solid: (θ1 , φ1 ) = (0, 0), (θ2 , φ2 ) = (π, 0), (θ3 , φ3 ) = (π/2, 0),
(θ4 , φ4 ) = (π/2, π); Solid: (θ1 , φ1 ) = (0, 0), (θ2 , φ2 ) = (π/3, 0),
(θ3 , φ3 ) = (2π/3, 0), (θ4 , φ4 ) = (π, 0); Dotted: (θ1 , φ1 ) = (0, 0),
(θ2 , φ2 ) = (2π/3, 0), (θ3 , φ3 ) = (π/2, π), (θ4 , φ4 ) = (π/3, π/2). The marked
points are computed from finite element package COMSOL.
GWU – p.32
Diffusion on the Surface of a Sphere: X
For N → ∞, the optimal energy for elliptic Fekete points gives
4
1
1
N 2 + N log N + l1 N + l2 , N → ∞ ,
max p(x1 , . . . , xN ) ∼ log
4
e
4
with l1 = 0.02642 and l2 = 0.1382.
E. A. Rakhmanov, E. B. Saff, Y. M. Zhou, “Electrons on the
Sphere”, in: Computational Methods and Function Theory 1994 (Penang),
293–309 and B. Bergersen, D. Boal, P. Palffy-Muhoray, “Equilibrium
Configurations of Particles on the Sphere: The Case of Logarithmic
Interactions”, J. Phys. A: Math Gen., 27, No. 7, (1994), pp. 2579–2586.
Reference:
This yields a key scaling law for the mininum of the averaged MFPT as
Principal Result:
For N 1, and N circular disks of common radius εa, and
with small area fraction N ε2 a2 1 with |S| = 4π, then
#
"
!
PN
1
j=1 |Ωεj |
− log
− 4l1 − log 4 + O(N −1 ) .
min ū ∼
ND
|S|
GWU – p.33
Diffusion on the Surface of a Sphere: XI
Estimate, with physical parameters, the minimum time taken
for a surface-bound molecule to reach a molecular cluster on a spherical
cell.
Application:
The diffusion coefficient of a typical surface molecule
(e.g. LAT) is ≈ 0.25µm2 /s and consider N = 100 signaling regions (traps)
of radius 10nm on a cell of radius 5µm. With these parameters,
Physical Parameters:
ε = 0.002 ,
N πε2 /(4π) = 0.01 .
Use scaling law to get asymptotic lower bound on the
averaged MFPT. For N = 100 traps, the bound is 7.7s, achieved at the
elliptic Fekete points.
Scaling Law:
As a comparison, for one big trap of the same area the
averaged MFPT is 360s, which is very different.
One Big Trap:
Conclusion: Both the Spatial Distribution and Fragmentation Effect of
Localized Traps are Rather Signficant at Moderately Small Values of ε.
GWU – p.34
Narrow Escape Problem Revisited
Narrow Escape Problem
for MFPT v(x) and averaged MFPT v̄:
1
∆v = − , x ∈ Ω ,
D
∂n v = 0 x ∈ ∂Ωr ; v = 0 ,
x ∈ ∂Ωa = ∪N
j=1 ∂Ωεj .
What is effect of spatial arrangement of traps on the
boundary in 2-D and 3-D? Need a higher order asymptotic theory.
Key Question:
S. Pillay, M.J. Ward, A. Pierce, R. Straube, T. Kolokolnikov, An
Asymptotic Analysis of the Mean First Passage Time for Narrow Escape
Problems, submitted, SIAM J. Multiscale Modeling, (2009).
Reference:
GWU – p.35
Narrow Escape From a Sphere: I
The surface Neumann G-function, Gs , is central:
4Gs =
Lemma:
1
,
|Ω|
x ∈ Ω;
∂r Gs = δ(cos θ − cos θj )δ(φ − φj ) ,
Let cos γ = x · xj and
x ∈ ∂Ω ,
Gs dx = 0 . Then Gs = Gs (x; xj ) is
7
2
1
1
1
−
+
(|x|2 + 1) +
log
.
Gs =
2π|x − xj | 8π
4π
1 − |x| cos γ + |x − xj |
10π
R
Ω
9
Define the matrix Gs using R = − 20π
and Gsij ≡ Gs (xi ; xj ) as


R
Gs12
···
Gs1N


R
···
Gs2N 
 Gs21
,
Gs ≡ 
..
..
..
..


.


.
.
.
R
GsN 1 · · · GsN,N −1
Remark:
As x → xj , Gs has a subdominant logarithmic singularity:
1
1
Gs (x; xj ) ∼
−
log |x − xj | + O(1) .
2π|x − xj | 4π
GWU – p.36
Narrow Escape From a Sphere: II
Principal Result:
For ε → 0, and for N circular traps of radii εaj centered at
xj , for j = 1, . . . , N , the averaged MFPT v̄ satisfies
"
PN 2
2
2πε
|Ω|
j=1 cj
1 + εlog
+
pc (x1 , . . . , xN )
v̄ =
2πεDN c̄
ε
2N c̄
N c̄

N
ε X
−
cj κj + O(ε2 log ε) .
N c̄ j=1
Here cj = 2aj /π is the capacitance of the j th circular absorbing window of
radius εaj , c̄ ≡ N −1 (c1 + . . . + cN ), |Ω| = 4π/3, and κj is defined by
3
cj
2 log 2 − + log aj .
κj =
2
2
Moreover, pc (x1 , . . . , xN ) is a quadratic form in terms C t = (c1 , . . . , cN )
pc (x1 , . . . , xN ) ≡ C t Gs C .
1) A similar result holds for non-circular traps. 2) The logarithmic
term in ε arises from the subdominant singularity in Gs .
Remarks:
GWU – p.37
Narrow Escape From a Sphere: III
Let N = 1, c1 = 2/π, and a1 = 1, (compare with Holcman..)
2
9
3
ε
ε
|Ω|
− − 2 log 2 +
+ O(ε2 log ε) .
1 + log
+
v̄ =
4εD
π
ε
π
5
2
One Trap:
of common radius ε:
ε
9N
2
ε
|Ω|
+
−
[1+ log
+ 2(N − 2) log 2
v̄ =
4εDN
π
ε
π
5
4
3
2
+ + H(x1 , . . . , xN ) +O(ε log ε) ,
2 N
N Identical Circular Traps:
with discrete energy H(x1 , . . . , xN ) given by
H(x1 , . . . , xN ) =
N X
N X
i=1
k=1
1
1
1
− log |xi − xk | − log (2 + |xi − xk |)
|xi − xk | 2
2
k6=i
.
Minimizing v̄ corresponds to minimizing H. This discrete
energy is a generalization of purely Coulombic or logarithmic energies
leading to Fekete points.
Key point:
GWU – p.38
Narrow Escape From a Sphere: IV
4
20
16
12
v̄
4
4
4
4
4
4
4
4
4
0.1
0.2
8
4
0
0.0
4
4
4
0.3
0.4
0.5
v̄ vs. ε with D = 1 and either N = 1, 2, 4 equidistantly spaced circular
windows of radius ε. Solid: 3-term expansion. Dotted: 2-term expansion.
Discrete: COMSOL. Top: N = 1. Middle: N = 2. Bottom: N = 4.
N =1
N =2
N =4
ε
v̄2
v̄3
v̄n
v̄2
v̄3
v̄n
v̄2
v̄3
v̄n
0.02 53.89 53.33 52.81 26.95 26.42 26.12 13.47 13.11 12.99
5.18
5.12
0.05 22.17 21.61 21.35 11.09 10.56 10.43 5.54
0.10 11.47 10.91 10.78 5.74
5.21
5.14
2.87
2.51
2.47
0.20 6.00
5.44
5.36
3.00
2.47
2.44
1.50
1.14
1.13
0.50 2.56
1.99
1.96
1.28
0.75
0.70
0.64
0.28
0.30
Plot:
GWU – p.39
Narrow Escape From a Sphere: V
2.0
1.6
1.2
v̄
0.8
0.4
0.0
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
v̄(ε) for D = 1, N = 11, and three trap configurations. Heavy: global
minimum of H (right figure). Solid: equidistant points on equator. Dotted:
random.
Plot:
Table: v̄ agrees well with COMSOL even at ε = 0.5. For ε = 0.5 and
N = 4, absorbing windows occupy ≈ 20% of the surface. Still, the
3-term asymptotics for v̄ differs from COMSOL by only ≈ 10%.
For ε = 0.1907, N = 11 traps occupy ≈ 10% of surface area; optimal
arrangement gives v̄ ≈ 0.368. For a single large trap with a 10%
surface area, v̄ ≈ 1.48; a result 3 times larger.
GWU – p.40
Narrow Escape From a Sphere: VI
spatial arrangement and fragmentation of traps on the sphere
is a very significant factor for v̄
Conclusion:
Key Ingredients in Derivation of Main Result:
The Neumann G-function has a subdominant logarithmic singularity on
the boundary (related to surface diffusion)
Tangential-normal coordinate system used near each trap.
Asymptotic expansion of global (outer) solution and local (inner
solutions near each trap.
Leading-order local solution is electrified disk problem in a half-space,
with capacitance cj .
Logarithmic switchback terms in ε needed in global solution
(ubiquitous in Low Reynolds number flow problems)
Need corrections to the tangent plane approximation in the inner
region, i.e. near the trap. This determines κj .
Asymptotic matching and solvability conditions (Divergence theorem)
determine v and v̄
GWU – p.41
Narrow Escape From a Sphere: VII
Numerical Computations:
classic energies
N X
N
X
to compare optimal points of H with those of
1
,
HC =
|xi − xj |
i=1 j=i+1
Hlog = −
(preliminary work with A. Cheviakov, MJW).
N X
N
X
log |xi − xj |.
i=1 j=i+1
Numerical Methods:
(cf. G. Beliakov, Optimization
Methods and Software, 19 (2), (2004), pp. 137-151).
Extended Cutting Angle method (ECAM).
Dynamical systems – based optimization (DSO).
Rubinov, and J. Yearwood, (2005)).
(cf. M.A. Mammadov, A.
Results:
For N = 5, 6, 8, 9, 10 and 12, optimal point arrangments coincide
Some differences for N = 7, 11, 16.
GWU – p.42
N=7:
Narrow Escape From a Sphere: VIII
Left: H. Right: Hc and Hlog .
N=11:
Left: H. Middle: Hc . Right: Hlog .
N=16:
Left: H and Hlog . Right: Hc .
GWU – p.43
Narrow Escape From a Sphere: IX
holds for arbitrary-shaped traps with two changes. Let Ωj be
the trap magnified by O(ε−1 ). (possibly multi-connected to allow for
receptor clustering).
Main Result:
Capacitance: cj
is now determined from
Lwc = 0 ,
∂ η wc = 0 ,
η ≥ 0,
on η = 0 , (s1 , s2 ) ∈
/ Ωj ;
wc ∼ cj /ρ ,
Correction to Tangent Plane:
on η = 0 , (s1 , s2 ) ∈ Ωj ,
wc = 1 ,
as ρ → ∞ .
κj now determined from
w2hηη + w2hs1 s1 + w2hs2 s2 = 0 ,
∂η w2h = 0 ,
−∞ < s1 , s2 < ∞ ,
η = 0 , (s1 , s2 ) ∈
/ Ωj ;
w2h ∼ −κj cj /ρ ,
η ≥ 0,
−∞ < s1 , s2 < ∞ ,
w2h = −K(s1 , s2 ) ,
η = 0 , (s1 , s2 ) ∈ Ωj ,
as ρ = (η 2 + s21 + s22 )1/2 → ∞ ,
where K(s1 , s2 ) is defined from
K(s1 , s2 ) = −
1
4π
Z
log |s̃ − s| wcη |η=0 ds .
Ωj
GWU – p.44
Further Directions
Narrow escape problems in arbitrary 3-d domains: require Neumann
G-functions with boundary singularity
Surface diffusion on arbitrary 2-d surfaces: require Neumann
G-function and regular part.
Include chemical reactions occurring within each trap, with detailed
mechanism of escape from trap through binding and unbinding events.
Can diffusive transport between traps influence stability of steady-state
of time-dependent localized reactions (ode’s) valid inside each trap?
Formulation leads to a Steklov-type eigenvalue problem.
Pattern formation for reaction-diffusion systems with localized spots on
curved and evolving surfaces.
Schnakenburg model on a Manifold: S. Ruuth (JCP, 2008)
GWU – p.45
Available at:
References
http://www.math.ubc.ca/ ward/prepr.html
T. Kolokolnikov, M. Titcombe, MJW, Optimizing the Fundamental
Neumann Eigenvalue for the Laplacian in a Domain with Small Traps,
EJAM Vol. 16, No. 2, (2005), pp. 161-200.
R. Straube, M. J. Ward, M. Falcke, Reaction Rate of Small Diffusing
Moelecules on a Cylindrical Membrane, J. Statistical Physics, Vol. 129,
No. 2, (2007), pp. 377-405.
P.C. Bressloff, B.A, Earnshaw, M.J. Ward, Diffusion of Protein
Receptors on a Cylindrical Dendritic Membrane with Partially
Absorbing Traps, SIAM J. Appl. Math., Vol. 68, No.5, (2008)
D. Coombs, R. Straube, M.J. Ward, Diffusion on a Sphere with
Localized Traps: Mean First Passage Time, Eigenvalue Asymptotics,
and Fekete Points, to appear, SIAM J. Appl. Math., (2009),
S. Pillay, M.J. Ward, A. Pierce, R. Straube, T. Kolokolnikov, An
Asymptotic Analysis of the Mean First Passage Time for Narrow
Escape Problems, submitted, SIAM J. Multiscale Modeling, (2009).
R. Straube, M. J. Ward, Intraceulluar Signalling Gradients Arising from
Multiple Compartments: A Matched Asymptotic Expansion Approach,
accepted, SIAM J. Appl. Math, (2009).
GWU – p.46
Narrow Escape in 2-D: I
Consider the narrow escape problem
from a 2-D domain. The surface
R
Neumann G-function, G, with Ω G dx = 0 is key:
1
, x ∈ Ω ; ∂n G = 0 , x ∈ ∂Ω\{xj } ,
|Ω|
1
G(x; xj ) ∼ − log |x − xj | + R(xj ; xj ) , as x → xj ∈ ∂Ω ,
π
4G =
Then define the Green’s function matrix

···
R1 G12

···
 G21 R2

G≡ .
..
..
.
.
 .
.
GN 1 · · · GN,N −1
G1N
G2N
..
.
RN



.


The local or inner problem near the j th arc determines a constant dj
w0ηη + w0ss = 0 ,
∂ η w0 = 0 ,
on |s| > lj /2 ,
0 < η < ∞,
η = 0;
w0 ∼ [log |y| − log dj + o(1)] ,
−∞ < s < ∞ ,
w0 = 0 ,
on |s| < lj /2 ,
as |y| → ∞ ,
dj = lj /4 .
η = 0.
GWU – p.47
Narrow Escape in 2-D: II
Principal Result:
Consider N well-separated absorbing arcs of length εlj for
j = 1, . . . , N centered at xj ∈ ∂Ω. Then, in the outer region
|x − xj | O(ε) for j = 1, . . . , N the MFPT is
v ∼ −π
N
X
Ai G(x; xi ) + χ ,
χ = v̄ =
i=1
1
|Ω|
Z
v dx ,
Ω
where a two-term expansion for Aj and χ are
Aj ∼
|Ω|µj
N Dπ µ̄
v̄ ≡ χ ∼
1−π
N
X
µi Gij +
i=1
π
pw (x1 , . . . , xN )
N µ̄
!
+ O(|µ|2 ) ,
|Ω|
|Ω|
+ 2
pw (x1 , . . . , xN ) + O(|µ|) .
2
N Dπ µ̄ N Dµ̄
Here pw is a weighted discrete sum in terms of Gij :
pw (x1 , . . . , xN ) ≡
N
N X
X
i=1 j=1
Remark:
µi µj Gij ,
µj = −
1
,
log(εdj )
dj =
lj
.
4
there is an analogous result that sums all logarithmic terms for v̄.
GWU – p.48
For N
Narrow Escape in 2-D: III
= 1 arc of length |∂Ωε1 | = 2ε (i.e. d = 1/2), then
ε
i
|Ω| h
− log
+ π (R(x1 ; x1 ) − G(x; x1 )) ,
v(x) ∼
Dπ
2
ε
i
|Ω| h
− log
+ πR(x1 ; x1 ) .
v̄ = χ ∼
Dπ
2
Extension of work of Singer et al. to arbitrary Ω with smooth ∂Ω.
For the unit disk, G and R are
|x|2
1
1
−
,
G(x; ξ) = − log |x − ξ| +
π
4π
8π
R(ξ; ξ) =
1
.
8π
For N equidistant arcs on unit disk, i.e. xj = e2πij/N for j = 1, . . . , N ,


N
X
N
εN
1 
+
G(x; xj ) ,
−π
− log
v(x) ∼
DN
2
8
j=1
εN
N
1
− log
+
,
χ∼
DN
2
8
GWU – p.49
Narrow Escape in 2-D: IV
Key Point: Spatial Arrangement of Arcs is Very Significant
2.0
1.5
χ
1.0
0.5
0.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Comparison of the two-term result for χ (dotted curves) with the
log-summed result (solid curves) vs.ε for D = 1 and for four traps on the
boundary of the unit disk. Trap locations at x1 = eπi/6 , x2 = eπi/3 ,
x3 = e2πi/3 , x4 = e5πi/6 (top curves); x1 = (1, 0), x2 = eπi/3 , x3 = e2πi/3 ,
x4 = (−1, 0) (middle curves); x1 = eπi/4 , x2 = e3πi/4 , x3 = e5πi/4 ,
x4 = e7πi/4 (bottom curves).
Plot:
GWU – p.50
Narrow Escape in 2-D: V
For one absorbing arc of length 2ε on a smooth boundary,
ε
i
|Ω| h
− log
+ πR(x1 ; x1 ) .
v̄ = χ ∼
Dπ
2
2 2
π
πµ
1
µ1
1
∗
3
−
R(x1 ; x1 ) + O(µ1 ) , µ1 ≡ −
λ(ε) ∼ λ ∼
|Ω|
|Ω|
log[ε/2]
Optimization:
Principal Result:
The maxima (minima) of R(x0 , x0 ) do not necessarily
coincide with the maxima (minima) of the curvature κ(θ) of the boundary
of a smooth perturbation of the unit disk. Consequently, for ε → 0, λ(ε)
does not necessarily have a local minimum (maximum) at the location of a
local maximum (minimum) of the curvature of a smooth boundary.
based on explicit perturbation formula for R(x0 , x0 ) for arbitrary
smooth perturbations of the unit disk.
Proof:
GWU – p.51